Reports of the Department of Mathematical Information Technology
Series B. Scientific Computing
No. B. 4/2013
Uncertainties in the heat conduction problems
and reliable estimates
Nikolay Banichuk
Pekka Neittaanmäki
Svetlana Ivanova
Tero Tuovinen
University of Jyväskylä
Department of Mathematical Information Technology
P.O. Box 35 (Agora)
FI–40014 University of Jyväskylä
FINLAND
fax +358 14 260 2731
http://www.mit.jyu.fi/
c 2013
Copyright Nikolay Banichuk and Svetlana Ivanova and Pekka Neittaanmäki
and Tero Tuovinen
and University of Jyväskylä
ISBN 978-951-39-5194-8(nid.)
ISBN 978-951-39-6938-7 (PDF)
ISSN 1456-436X
Uncertainties in the heat conduction problems
and reliable estimates∗
Nikolay Banichuk
Svetlana Ivanova
Tero Tuovinen
Pekka Neittaanmäki
April 24, 2013
Abstract
The heat conduction problems for anisotropic bodies are studied taking into
account the uncertainties in the material orientation. The best estimations of
the upper and lower bounds of the considered energy dissipation functional
are based on the developing new approach consisting in solution of some optimization problems and finding the extremal internal material structures, which
realize minimal and maximal dissipation.
1
Introduction
The problems of incompleteness of data and uncertainties are typical for anisotropic
solids and structures having chaotic orientation of small material particles such as
grains, crystal or short nano-fibers. Different possible compositions of elementary
particles with various orientations result in different values of such integral characteristics as a total dissipation energy in the heat conduction problems, total potential
energy in the thermoelasticity and thermoconductivity problems. Taking into account the conditions of uncertainties concerning the material orientations it is very
important to obtain various estimations of the considered functionals and in particular limiting estimates known as double-side or bilateral estimates Banichuk and
Neittaanmäki (2010).
In the proposed article the problem of estimation of dissipation energy characteristics is considered for anisotropic body constituting of the locally orthotropic
material. It is assumed that an orientation of the principle axes of orthotropy is not
known beforehand at each point of the body and can be distributed by various ways
in different parts of the body including chaotic orientation. The search for doubleside estimates is reduced to the solution of optimization problems and finding the
extremal orientations of the orthotropy axes.
∗
This research was supported by the Academy of Finland (grant no. 140221 and grant no. 269351);
by the Program for RFBR (Grant 11-08-00030-a); by the RAS program 12, Program of Supports of
Leading Scientific Schools (grant 2611.2012.1)
1
2
Heat conduction problem for bodies from locally orthotropic material
Let us consider heat conduction problem for solid body occupied the domain Ω (see
Figure 1) with the boundary Γ = Γg + Γi (Γg ∩ Γi = 0). The material of the body
is anisotropic with respect to the heat conduction process described by the known
relations Landau and Lifshitz (1965), Berdichevsky (2009), Nowacki (1970)
ϕ = θ−1
q = D · ∇ϕ,
(1)
where θ is a temperature, q is a vector of the heat flux and D is a heat conduction
tensor of the second rank. In the case of absence of the source of heat in the domain Ω
we will use the boundary conditions, governing equation and the quality functional
(dissipation of energy) in the following form
(ϕ)Γg = ϕ0 ,
(n · D · ∇ϕ)Γi = 0
∇ · (D · ∇ϕ) = 0
x∈Ω
ˆ
J=
∇ϕ · D · ∇ϕdΩ
(2)
(3)
(4)
Ω
where ϕ0 is a given function specified on Γg , n is an outwards unit normal vector
specified on the part Γi , (·) between the vectors means the scalar product and the
symbol ∇ is the gradient operator, i.e.
∂ϕ ∂ϕ ∂ϕ
∇ϕ =
,
,
.
∂x1 ∂x2 ∂x3
In accordance with the variational principle Berdichevsky (2009) the actual distribution of the function ϕ realizes a minimum for the functional J on the set of admissible functions satisfying the first boundary condition in (2), if
J → min
(5)
ϕ
Note the second boundary condition in the (2) plays the role of transversality condition for the functional (4) and is satisfied ”automatically” for extremum solution.
Note that the equation (3) is the Eulerian equation for the functional (4).
In what follows we will suppose that the material is locally orthotropic and the
orientation of the axes of orthotropicity is unknown beforehand. Let us fix the unit
vectors e01 , e02 , e03 of orthogonal coordinate system x1 , x2 , x3 which is considered as a
global system (see Figure 2). The principal directions unit vectors e1 , e2 , e3 of the
heat conduction tensor D of orthotropic material (axes of local symmetry) at the
arbitrary point (x1 , x2 , x3 ) ∈ Ω are related with the global coordinate vectors e01 , e02 , e03
by means of the rotation tensor Q = Q(x) as
ei = Q ∗ e0i = Q · e0i
2
(i = 1, 2, 3)
(6)
Γ
Γi
Figure 1: Domain Ω with given temperature Γi (black) and thermally insulated Γg
(grey) boundary conditions.
Figure 2: Transformation of global unit vectors to the local material principle vectors
by rotation tensor Q.
3
QT · Q = Q · QT = E
(7)
where the symbol T means the operation of transposition and E = {δij } - unit tensor,
where δij is a Kronecker symbol (i, j = 1, 2, 3) and (∗) is a tensor operation of the
rotation. In the axes of symmetry of the orthotropic material the heat conduction
tensor D is written as
0 0
0
(8)
ei ⊗ e0j = Q ∗ D0
Q · e0i ⊗ Q · e0j = Q ∗ Dij
D = Dij ei ⊗ ej = Dij
where ⊗ is tensor product and
0
· e0i ⊗ e0j
D0 = Dij
(9)
The expression for the heat conduction tensor D can be rewritten in the form
0 0
0
Q · e0i ⊗ Q · e0j = Q · Dij
ei ⊗ e0j · QT = Q · D0 · QT
(10)
D = Dij
If κi and e0i i.e. eigenvalues and eigenvectors of the tensor D0 , i.e.
D0 · e0i = κi0 e0i
(11)
then κi0 and ei = Q · e0i i.e. eigenvalues and eigenvectors of the tensor D = Q ∗ D0 ,
i.e.
D · ei = κi0 ei
(12)
Taking into account the equations (7), (8) and (11) we will have the equation (12). In
fact
D · ei = Q ∗ D0 · Q ∗ e0i = Q ∗ D0 · e0i = κi0 Q ∗ e0i = κi0 ei
(13)
For given tensor D0 the values of the functionals J depend on the realization of
Q = Q(x) and corresponding actual values of ϕ = ϕ(x), minimizing the functional
of energy dissipation (4) under constraints (2) (for considered Q(x), i.e.
J (Q, ϕ∗ ) = min J (Q, ϕ)
ϕ
3
(14)
Uncertainties in orientation of orthotropic material
and double-side estimates
If there is no data concerning material orientation, i.e. the tensor-function Q = Q(x)
(x ∈ Ω), characterizing materia distribution is unknown, then it is very important to
obtain the lower and upper bounds of J, i.e. to find the limit double-side estimates
Jmin and Jmax , such that
Jmin ≤ J (Q, ϕ∗ ) ≤ Jmax
(15)
for any realization of Q satisfying the condition (7).
To obtain reliable estimations of the dissipation energy functional J and other
important characteristics we apply in the paper an approach based on the solution
4
of two optimization problems. The following problem is devoted to finding the
lower estimate
Jmin = min J (Q, ϕ∗ ) = min min J (Q, ϕ)
(16)
Q
ϕ
Q
and another problem consists in the searching of the upper bounds
Jmax = max J (Q, ϕ∗ ) = max min J (Q, ϕ)
Q
Q
ϕ
(17)
where min and max with respect to Q in the equations (16) and (17) are determined
under constraint (7). Operation min with respect to ϕ in the equations (16) and (17)
is performed taking into account boundary conditions from the equation (2).
In what follows we will study the proposed approach and problems of searching
the extremum of J with respect to Q
J → extr
(18)
Q
and analyze extremum conditions and behavior equations.
4
Extremal conditions for orthotropic material orientation
To derive extremum conditions defining the orthogonal tensor of rotation Q = Q(x)
and characterizing the extremal orientations of orthotropy axes let us use the method
of Lagrange multipliers and construct augmented functional
J L = J + JP
(19)
P · · QT · Q − E dΩ
(20)
ˆ
JP =
Ω
ˆ
∇ϕ · Q ∗ Q
J=
0
ˆ
∇ϕ · Q · D0 · QT · ∇ϕdΩ
· ∇ϕdΩ =
Ω
(21)
Ω
where (··) between tensors mean double scalar product and symmetric tensor of
second rank P = P (x) (x ∈ Ω) is Lagrange multiplier, specifying in Ω and corresponding to the condition of orthogonality (see equation (7)). The dissipation energy functional J can be also rewritten as
ˆ
J=
B · · Q · D0 · QT dΩ
(22)
Ω
Here by means of B we denote the following symmetric second rank tensor
B = ∇ϕ ⊗ ∇ϕ,
and the symbol ⊗ is the tensor product.
5
BT = B
(23)
Let us derive the following expressions for the first variations δJ and δJP with
respect to variation δQ pf rotation tensor Q. We will have
ˆ
ˆ
0
T
0
T
δJ =
B · · δQ · D · Q + Q · D · δQ dΩ = 2 δQ · · D0 · QT · B dΩ (24)
Ω
Ω
ˆ
ˆ
and
T
T
δQ · · P · QT dΩ
P · · δQ · Q + Q · δQ dΩ = 2
δJP =
Ω
(25)
Ω
Taking into account the expressions (19) – (21), (24) and (25) we will find the expression for the total variation δQ in the following form
ˆ
L
(26)
δJ = δJ + δJP = 2 δQ · · D0 · QT · B + P · QT dΩ
Ω
Using the extremum condition
δJ L = 0
(27)
and arbitrariness of Q, i.e. arbitrariness of δQ, we will have
D0 · QT · B + P · QT = 0,
x∈Ω
(28)
Multiplying the relation (28) on Q and using formulae (10) and (23) we find
D · ∇ϕ ⊗ ∇ϕ = −Q · P · QT ,
x∈Ω
(29)
This relation means the symmetry of the second rank tensor
(D · ∇ϕ) ⊗ ∇ϕ
written in the left hand side of the equality (29), i.e.
(D · ∇ϕ) ⊗ ∇ϕ = ∇ϕ ⊗ (D · ∇ϕ)
(30)
The equality (30) is satisfied if the vectors D · ∇ϕ and ∇ϕ are parallel, i.e.
D · ∇ϕ = λ∇ϕ
(31)
where λ is some scalar value.
5
Double-side estimates based on derived extremal conditions
The necessary extremum condition (31) for dissipation energy functional J with respect to rotation tensor Q, defining an extremal distribution of Q aand expressing
the collinearity of the vectors ∇ϕ and
D · ∇ϕ = Q · D · QT · ∇ϕ
6
is an eigenvalue problem. Consequently, the vector ∇ϕ is one of the eigenvectors of
the heat conduction tensor D:
D · ∇ϕ = λi ∇ϕ,
(32)
i = 1, 2, 3
Taking into account that the eigenvalues λi of the tensors D and D0 are equal (see
the equations (11) and (13)) and given, we assume
(33)
λ1 = λmin < λ2 < λ3 = λmax
Substituting (32) into the Euler equation (3) of the functional J we obtain the equations that determine the stationay distribution of scalar function ϕ = ϕ(x) :
∇ · (λi ∇ϕ) = 0,
x∈Ω
(i = 1, 2, 3),
in the case of specified rotation tensor Q according to the equation
Q · D0 · QT · ∇ϕ = λi ∇ϕ
(34)
(35)
The elliptical partial differential equation (34) with the boundary conditions
(ϕ)Γg = ϕ0 ,
(λi n · ∇ϕ)Γi = 0
(36)
corresponding to conditions (2) with the relations (32) constitute the conventional
boundary value problem describing, as it is well known, homogeneous or nonhomogeneous isotropic processes of the heat conductivity. Under some known additional constraints superimposed on the boundary shape Γ = Γg + Γi (Γg ∩ Γi = 0)
we have the existence and uniqueness of the solution of (34) and (36) with given λi .
If we assume that the same way of extremum orientation of the principle axes of
orthotropy is realized for all domain Ω, then λi is constant in Ω and the considered
heat conduction process is described by the classical boundary value problem
x∈Ω
∆ϕ = 0,
(ϕ)Γg = ϕ0 ,
(37)
(n · ϕ)Γi = 0
(38)
for Laplace equation with mixed (in general case) boundary conditions. Here ∆ is a
Laplace operator acting in 3-dimensional space.
Note that the equality in the equation (37) means that in the case of the body with
extremum orthotropy the heat conduction process is described by the same equation
as in the isotropic case. If the domain Ω consists of several sub-domains Ωi such that
Ω = ∪Ωi ,
Ωi ∩ Ωj = 0
(i 6= j)
(39)
and for each separate sub-domain Ωi the same extremum way of material orientation is taken, then the isotropic heat conduction process is realized for all considered
subdomains.
Let us assume that the orthotropic material is distribute in accordance with the
same extremum rule in the domain Ω. Then we will have the ”isotropic” boundary
7
value problem (37) and (38), and consequently the state variable ϕ (inverse temperature) is independent of λi . As a result we obtain the following minimal and maximal
values of the considered quality functional J:
min J = λmin I
(40)
max J = λmax I
(41)
Q
Q
ˆ
where
(∇ϕ)2 dΩ
I=
(42)
Ω
Thus the double-side estimates of the energy dissipation functionals can be written as
J
(43)
λmin ≤ ≤ λmax
I
6
Two-dimensional case of extremal material orientation
Separately consider the two-dimensional case with plane domain Ω. In this case
∂ϕ ∂ϕ
,
,
x = {x1 , x2 } ∈ Ω
(44)
∇ϕ =
∂x1 ∂x2
Then the element of orthogonal tensor Q are represented in the form
Q21 = −Q12 = sin α
Q11 = Q22 = cos α
(45)
where α is the angle of rotation of the specified tensor Q. On the basis of the equation (35) we obtain an explicit expression relating the angle α = α(x1 , x2 ) with the
function ϕ = ϕ(x1 , x2 ). For definiteness assume that the vector ∇ϕ, presented in the
equation (44), correspond to the eigenvalue λi . Then the eigenvector k, corresponding to the eigenvalue λj (i 6= j) is
∂ϕ
∂ϕ
,−
(46)
k=
∂x2 ∂x1
which is orthogonal to the eigenvector ∇ϕ from the equation (44). We form a scalar
product of both sides of the vector equality (32) with the vector k. We will have
k · D · ∇ϕ = 0
(47)
This relation contains two separate cases. The first case:
cos 2α = C,
sin 2α = S
8
(48)
where
(
2 2 )
∂ϕ
∂ϕ
0 ∂ϕ ∂ϕ
0
0
−
+ 4D12
(D11
− D22
)
∂x1
∂x2
∂x1 ∂x2
q
C=−
0 2
0 2
0
)
) + 4 (D12
− D22
(∇ϕ)2 (D11
(49)
and
(
2 2 )
∂ϕ
∂ϕ
∂ϕ
∂ϕ
0
0
0
2 (D11
− D22
)
− 2D12
−
∂x1 ∂x2
∂x1
∂x2
q
S=
0 2
0 2
0
)
) + 4 (D12
− D22
(∇ϕ)2 (D11
(50)
corresponds to the smaller eigenvalue λ1 (λ1 < λ2 ). The second case:
cos 2α = −C,
sin 2α = −S
(51)
corresbonds to the larger eigenvalue λ2 (λ2 > λ1 ).
7
Examples of double-side estimates
Suppose at first that the orthotropic material occupies the 3-dimensional domain Ω
situated between the internal sphere of radius r1 , where r1 , r2 (r1 < r2 ) is given values. The temperature of θ = θ1 is defined at the internal boundary and the temperature θ = θ2 is given at the external boundary, where θ1 < θ2 . Note that θi , (i = 1, 2)
are given and positive values. Thus we consider the following boundary conditions
ϕ = ϕ1 =
1
,
θ1
r = r1
1
,
r = r2
(52)
θ2
where ϕ1 < ϕ2 . Here we use spherical coordinate system with the origin at r = 0.
From the properties of symmetry it follows that the extremum orientations of the
axes of orthotropy with
ϕ = ϕ2 =
λ1 = λmin
and
λ3 = λmax
and
J → max
corresponding respectively to the cases
J → min
Q
Q
are realized in radial direction. Besides, the gradient of ϕ, i.e. vector ∇ϕ, and also the
heat flux vector q are directed along the radius vector at each point of the domain
Ω. Note that the heat flux q is absent in circumferential directions. The following
values characterize the extremal distribution of material:
qmin = λmin N r0 ,
qmax = λmax N r0
9
(53)
x2 e2 e1 Ω e20 α e10 x1 Figure 3: Orientation of local orthotropicity in 2-dimensional case.
λmin I ≤ J ≤ λmax I
r
ϕ2 − ϕ1
,
r0 =
N=
r2 − r1
|r|
ˆ
where
I=
4
(∇ϕ)2 dΩ = πN (ϕ2 − ϕ1 ) r12 + r1 r2 + r22
3
Ω
(54)
and r0 is an unit vector, oriented in radial direction.
Next let us consider the problem of finding the double-side estimates when a
simply connected domain Ω occupied by the orthotropic material is a rectangular
parallelpiped with the upper and lower faces at x3 = −c and x3 = c and side faces
at x1 = ±a and x2 = ±b. We use Cartesian coordinate system (x1 , x2 , x3 ) and we assume that the temperature θ is given at the lower and upper faces and the sidefaces
are thermally insulated, i.e. the boundary conditions have the form:
ϕ = ϕ1 =
1
,
θ1
x3 = −c
and
ϕ = ϕ2 =
1
,
θ2
x3 = c
(55)
and
q · n = n · D · ∇ϕ = 0 at x1 = ±a, x2 = ±b
10
(56)
where θ1 > 0, θ2 > 0 and (θ1 > θ2 ). Extremal material distribution and corresponding heat conduction processes are characterized by the existenec of level surfaces
x3 is constant (−c < x3 < c ∈ Ω) with a constant distribution of variable ϕ (constant temperature θ). The gradient of ϕ is parallel to x-axis. Therefore the axes of
orthotropy with minimal eigenvalue λ = λmin (in the case J → minQ ) and with
maximal eigenvalue λ = λmax (in the case of J → maxQ ) are oriented in a parallel
way with respect to the axis x3 . Such orientation provides, respectively, either the
minimium or the maximum of dissipation. For considered problem we will have
qmin = λmin Φx03 ,
min J = λmin I,
ϕ2 − ϕ1
,
2c
Q
∇ϕ = Φx03 ,
ˆ
where
(∇ϕ)2 dΩ =
I=
(57)
max J = λmax I
Q
Φ=
q = λmax Φx03
Ω
x03 =
x3
|x3 |
2ab
(ϕ2 − ϕ1 )2
c
(58)
and x03 is an unit vector of the x3 -axis, obtained when the vector x3 is devided by its
length |x3 |.
8
Notes and conclusion
In hte case, when the coefficient Dij and the considered eigenvalues λi are independent of x = (x1 , x2 , x3 ), then the anisotropic behaviour equation is reduced to the
Laplace equation which describes the heat conduction of homogeneous isotropic
body. Since the theory of the heat conduction of isotropic homogeneous solids is
well developed and solution of the corresponding boundary value problem has
been found (analytically and numerically) for most problems of practical importance, then this reduction allows to consider the above problem of obtaining of
double-side estimates to be solved.
Taking into account the conditions of uncertainties concerning material orientations we obtain various estimations of the considered functionals and in particular limiting estimates known as double-side or bilateral estimates. The search of
double-side estimates as it was shown is reduced to the solution of optimization
problems and finding the extremal orientation of the orthotropy axes.
References
Banichuk, N. V. and Neittaanmäki, P. J. (2010). Structural Optimization with Uncertainties. Springer-Verlag, Dordrecht. ISBN 978-90-481-2517-3.
Berdichevsky, V. (2009). Variational principles of continuum mechanics. Springer.
11
Landau, D. and Lifshitz, E. M. (1965). Teoriya uprugosti (Theory of Elasticity, 2nd ed.).
Oxford.
Nowacki, W. (1970).
Warszawa.
Teoria sprezystosci.
12
Pantwawe Wydawnictwo Naukowe,